<p>Quality Control is a term often used within production and is referring to managing processes so they produce capable products. Within Quality Control, process capability index is a common measure to oversee processes. Safety Region Plots were introduced to do this graphically. In Albing & Vännman (2010) the concept of Safety Region Plots is expanded to incorporate an elliptical shape. The method of Elliptical Safety Region Plots assumes a normally distributed data. In this paper we are looking at the robustness of the Elliptical Safety Region Plots if we can assume a symmetrically, but non-normal, distribution. In the results we can conclude that an adjustment is required for symmetric, but non-normal, data if the method in Albing & Vännman (2010) is going to be used. An eventual adjustment is discussed in discussions. To easily be able to use the Elliptical Safety Region Plots mentioned in Albing & Vännman (2010) we have developed a program in RExcel.</p>

<p>This thesis consits of four papers dealing with distance based (non-Euclidean) tests for spatial clustering in inhomogeneous populations. </p><p>The density adjusted distance (DAD), which considers the underlying density, is defined in the first paper. The proposed distance can be used together with any of the old distance based methods developed for traditional homogeneous spatial patterns. </p><p>The test statistics in distance based tests can all be seen as a weighted sum of distance measures for distances between <i>n</i> cases with known co-ordinates. DAD based test statistics are developed and their performance is compared with the performance of previously suggested tests by simulation in the second paper. The tests are compared in different types of data set and for various kinds of clustering. It is shown that no test is the optimal choice for all alternative hypotheses and that the tests are unequally sensitive to the structure of the underlying data. Tests based on the DAD are often a good alternative. </p><p>Test statistics and graphical tools for the Empirical Distribution Function of DAD are developed and examined in the third paper. We show that the result of an EDF test combined with EDF plots provides more information about the possible nature of clustering in a sample than the result of a parametric test only. </p>

<p>The use of process capability indices in the industry is traditionally based on the assumptions that a sample from a process are observations on independently, identically and normally distributed random variables<i>, IIN</i>. However, all three assumptions are open to discussion and in this thesis, the estimation of the indices is studied when the assumption of independence is not fulfilled.</p><p>In five reports, the indices <i>C</i><i>pk</i> and <i>C</i><i>pm </i>are studied, and instead of random samples, samples are regarded as observations on a time series.</p><p>In the first four reports, each index is studied for either an <i>AR(1)</i> or an <i>MA(1)</i> process and the fifth report, both indices are studied for a general <i>ARMA(p,q</i>) process.</p><p>In all reports, alternatives to <i>C</i><i>pk</i><i> </i>and <i>C</i><i>pm</i><i> </i>are suggested as well as point and interval estimators for the suggested indices. The accuracy of interval estimators are evaluated through large Monte Carlo simulations and the difference between empirical coverage rates and nominal confidence limits are calculated.</p><p>It was found in all reports that a dependency among observations has a great impact on the coverage rates. The coverage rate difference depends on both the size of the autocorrelation and the type of time series model and for the original <i>C</i><i>pk</i> and <i>C</i><i>pm</i> the difference can be large. With the suggested alternative indices, however, the differences are always reduced and unless the autocorrelations are close to ±1, the sizes of differences are of little consequence.</p>

Für ein vertieftes Verständnis der sozialwissenschaftlichen Datenanalyse werden in der vorliegenden Arbeit die Deskriptivstatistik (beschreibende Statistik) und die Inferenzstatistik (schließende Statistik als Instrument zur Charakterisierung der Grundgesamtheit aufgrund einer Stichprobe) integriert dargestellt. Im Zentrum der Darstellung steht die Analyse von Zusammenhängen von Merkmalen in Stichprobe und Grundgesamtheit in Abhängigkeit vom erzielten Messniveau. Anhand verschiedener sozialwissenschaftlicher Fragestellungen werden daher die Messniveaus und Skalierungsverfahren diskutiert, die Verteilung interessierender Merkmale in Stichprobe und Grundgesamtheit, die Grundlagen der Wahrscheinlichkeitstheorie, Auswahlverfahren zur Konstruktion von Stichproben, Schätz- und Testverfahren sowie insbesondere die verschiedenen Konzepte für die Zusammenhangsanalyse von Merkmalen.

This thesis consits of four papers dealing with distance based (non-Euclidean) tests for spatial clustering in inhomogeneous populations. The density adjusted distance (DAD), which considers the underlying density, is defined in the first paper. The proposed distance can be used together with any of the old distance based methods developed for traditional homogeneous spatial patterns. The test statistics in distance based tests can all be seen as a weighted sum of distance measures for distances between n cases with known co-ordinates. DAD based test statistics are developed and their performance is compared with the performance of previously suggested tests by simulation in the second paper. The tests are compared in different types of data set and for various kinds of clustering. It is shown that no test is the optimal choice for all alternative hypotheses and that the tests are unequally sensitive to the structure of the underlying data. Tests based on the DAD are often a good alternative. Test statistics and graphical tools for the Empirical Distribution Function of DAD are developed and examined in the third paper. We show that the result of an EDF test combined with EDF plots provides more information about the possible nature of clustering in a sample than the result of a parametric test only.

The use of process capability indices in the industry is traditionally based on the assumptions that a sample from a process are observations on independently, identically and normally distributed random variables, IIN. However, all three assumptions are open to discussion and in this thesis, the estimation of the indices is studied when the assumption of independence is not fulfilled. In five reports, the indices Cpk and Cpm are studied, and instead of random samples, samples are regarded as observations on a time series. In the first four reports, each index is studied for either an AR(1) or an MA(1) process and the fifth report, both indices are studied for a general ARMA(p,q) process. In all reports, alternatives to Cpk and Cpm are suggested as well as point and interval estimators for the suggested indices. The accuracy of interval estimators are evaluated through large Monte Carlo simulations and the difference between empirical coverage rates and nominal confidence limits are calculated. It was found in all reports that a dependency among observations has a great impact on the coverage rates. The coverage rate difference depends on both the size of the autocorrelation and the type of time series model and for the original Cpk and Cpm the difference can be large. With the suggested alternative indices, however, the differences are always reduced and unless the autocorrelations are close to ±1, the sizes of differences are of little consequence.

Quality Control is a term often used within production and is referring to managing processes so they produce capable products. Within Quality Control, process capability index is a common measure to oversee processes. Safety Region Plots were introduced to do this graphically. In Albing &amp; Vännman (2010) the concept of Safety Region Plots is expanded to incorporate an elliptical shape. The method of Elliptical Safety Region Plots assumes a normally distributed data. In this paper we are looking at the robustness of the Elliptical Safety Region Plots if we can assume a symmetrically, but non-normal, distribution. In the results we can conclude that an adjustment is required for symmetric, but non-normal, data if the method in Albing &amp; Vännman (2010) is going to be used. An eventual adjustment is discussed in discussions. To easily be able to use the Elliptical Safety Region Plots mentioned in Albing &amp; Vännman (2010) we have developed a program in RExcel.

Denna uppsats har gjorts på uppdrag av Ålands statistik- och utredningsbyrå (ÅSUB) i syfte att förklara och prognostisera de åländska företagens omsättningsutveckling över tiden. I uppsatsen har jag undersökt om en tidigare framtagen modell för att förklara Ålands omsättningsindex kan användas för att förklara olika branschindex på Åland. Jag har även tagit fram två olika modeller på ledande indikatorer från Sverige och Finland med ARIMA-feltermer. De ledande indikatorerna är svenska och finska konsumenters förtroendeindex samt industrins förtroendeindex och svenska OMX 30 börsindex. Modellerna använder olika fördröjningar på dessa fem indikatorer som förklaringsvariabler. Modell 1 använder utvalda fördröjningar och Modell 2 använder glidande medelvärden på fördröjningsintervall. Jag har testat hur bra de två modellerna kan förklara och prognostisera Ålands omsättningsindex och de olika branschindexen. Båda modellerna förklarar Ålands omsättningsindex bra. Modell 2 är klart bättre än Modell 1 på att förklara de olika branschindexen. Prognosförmågan för båda modellerna är diskutabel. Modell 2 är bättre än Modell 1 på att prognostisera både Ålands omsättningsindex och dess branschindex. Jag har visat att det går att ta fram en bra modell för att förklara Ålands omsättningsindex och dess branschindex med hjälp av ledande indikatorer från Sverige och Finland.

Given a random sample from a multivariate probability distribution p, the maximum minimum parents and children algorithm locates the skeleton of the directed acyclic graph of a Bayesian network for p provided that there exists a faithful Bayesian network and that the dependence structure derived from data is the same as that of the underlying probability distribution. The aim of this thesis is to examine the consequences when one of these conditions is not fulfilled. There are some circumstances where the algorithm works well even if there does not exist a faithful Bayesian network, but there are others where the algorithm fails. The MMPC tests for conditional independence between the variables and assumes that if conditional independence is not rejected, then the conditional independence statement holds. There are situations where this procedure leads to conditional independence being accepted that contradict conditional dependence relations in the data. This leads to edges being removed from the skeleton that are necessary for representing the dependence structure of the data.

The aim of this paper is to present simulation methods for the pricing of American financial instruments. Three methods are presented. Each differs from the others in it's approach to the problem and the method of finding a solution. We illustrate the variety of possible approaches that can be adopted when dealing with this complicated problem. The results of using these algorithms are compared with examples found in literature on the subject. We try to identify the factors that influence price estimators and provide some new results about the properties and distributions of those estimators. We show that even a simple variance reduction technique has a positive effect for these algorithms. The purpose of this paper is to present the effectiveness of a simulation method in pricing American options. This is contrary to the opinion often stated in articles and monographs that the simulation approach is not adequate for the task. We provide an overview and comparison of earlier methods proposed and follow this with an extended discussion. This paper sets the foundations for further research into use of these algorithms for multidimensional problems, where they may offer a substantial advantage over deterministic methods.